Problem: Simplify the following expression and state the condition under which the simplification is valid. $x = \dfrac{r^2 - 4}{r + 2}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = r$ $ b = \sqrt{4} = 2$ So we can rewrite the expression as: $x = \dfrac{({r} + {2})({r} {-2})} {r + 2} $ We can divide the numerator and denominator by $(r + 2)$ on condition that $r \neq -2$ Therefore $x = r - 2; r \neq -2$